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Articles Cosmology Cosmological Baryon-Number Generation in Grand Unified Models (1981)
Cosmological Baryon-Number Generation in Grand Unified Models (1981) |
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Cosmology is potentially an important source of information on the baryon-number-density-nonconserving interactions expected in most grand unified gauge models. Any net baryon number density 
imposed as an initial condition on the universe should have been rapidly destroyed by any 
-nonconserving interactions. To account for the observed ratio of baryon number density to photon number density, 
a net baryon number must subsequently have been generated. This requires, in addition to 
nonconservation, the violation of 
and 
(and hence 
) invariance, along with departures from thermal equilibrium. (1) , (2) This Letter outlines the complete calculation of 
generation in specific grand unified models in the context of the standard hot big-bang model of the early universe. The method we present allows for the exact treatment of an arbitrary number of superheavy bosons as well as the presence of nonthermalizing modes. (3) We summarize results for several realistic SU(5) models. Many details and extensions are discussed by Harvey et al. (4) in another paper. (4)
We denote heavy bosons generically by 
and light fermions by 
The number density 
of a particle 
, and that of its antiparticle 
are parametrized by 
and 
The time development of these quantities is described by a set of coupled Boltzmann transport equations. For the heavy bosons these are (2) , (4)
where dots denote time derivatives and the expansion of the universe is accounted for through division by 
in the definitions of 
The first terms on the right-hand side of Eqs. (1a) and (1b) correspond to free decays of and 
with partial rates 
averaged over the decaying energy spectrum. The second terms account for back reactions in which the decay products interact to produce 
The equilibrium number density 
is obtained by integrating the exp
equilibrium Maxwell-Boltzmann phase-space density. In equilibrium, 
and 
the expansion of the universe produces deviations from equilibrium at temperatures 
The densities of fermion species develop according to
where 
denotes the number of particles of type 
in the state 
denotes the difference in branching ratios between the conjugate decays 
and 
divided by the full rate for decay; it vanishes if is conserved. The first part of the first term on the right-hand side of Eq. (2) therefore represents the production of an asymmetry in fermion number density as a result of -nonconserving decays of a symmetrical 
mixture. The second part causes asymmetries, 
between and to be transferred to the fermions when the 
decays. The third part gives a correction to the rate for inverse decays resulting from the deviation of the fermion number densities from their equilibrium value. The second term in Eq. (2) represents the production and destruction of fermions by two-to-two scattering processes. 
is the cross section for this scattering mediated by exchange, but with the term corresponding to a real intermediate removed (since this is already accounted for by decay and inverse decay processes).
The number of independent particle densities to be treated in Eqs. (1) and (2), may be reduced by using unbroken symmetries (gauge and global). For non-Abelian groups, any asymmetries are shared symmetrically among members of each irreducible representation. If only a subset of the interactions that may potentially contribute to Eq. (2) are included there may be additional symmetries leading to further conserved combinations of fermion number densities (e.g., 
conservation in the absence of Higgs-fermion couplings for the models discussed below).
Let 
be the independent fermion asymmetries and 
the independent supermassive boson asymmetries. It is convenient to form a set 
which consists of independent quantum number densities 
etc. related to 
by a unitary transformation, 
The thermalization of a quantum number 
through reactions of a particular boson is given from Eq. (2) by 
where
and 
represents the change in the value of through the reaction 
Boltzmann's 
theorem requires that the eigenvalues of 
are all real and nonpositive. Any zero eigenvalues reveal additional symmetries; the corresponding eigenvector of number densities is then conserved in reactions [e.g., in vector boson exchanges in SU(5)].
We consider two grand unified models based on SU(5). In each case a family of fermions transforms as a reducible representation 
, labeled by the family index 
The following Higgs representations are taken to couple to fermions: in model I [ minimal SU(5)], a single 
of Higgs, 
in model II, 
and an additional of Higgs, 
The Yukawa couplings in these models have the schematic form 
It may be shown that a -nonconserving nonzero enters through an imaginary part of the product of the couplings in diagrams in which one boson is exchanged between the 
produced in the decay. The sum over 
and 
in Eq. (2) runs over all types and families of fermions; thus for fixed fermion types is proportional to a family-space trace of Yukawa coupling matrices. In model I the first diagram exhibiting nonconservation involves only Higgs bosons and is of eighth order in the Yukawa couplings. (4) , (5) , (6) It is proportional to the imaginary part of the family-space trace, Tr
suggesting the rough estimate 
where 
and 
is an averaged Yukawa coupling at unification scales. The naive expectation that will increase at low energy scales may be invalid if 
since the renormalization-group equation for will have positive and negative contributions of roughly equal size.
In model II, both and 
have only the single -nonconserving component, (7) (3, 1, 
); since is a complex representation one may form complex linear combinations so that the (3, 1, ) in both and 
is separately a mass eigenstate. This suffices to show that no nonconservation may occur for gauge boson decay with Higgs scalar exchange or vice versa. nonconservation may occur at 
through decay with exchange (and vice versa). (8)
SU(3) 
SU(2)
U(1)
symmetry allows the fifteen independent fermion fields in a family of an SU(5) model to be reduced to the set 
and 
(the subscript 
denotes the left-handed helicity state and denotes charge conjugation). The model contains a 
of -nonconserving vector bosons 
(with number densities parametrized by 
and 
). We consider the case where there are 
(=1 or 2) scalars, 
transforming as (3,1,) (with number densities parametrized by 
and 
These models possess a locally conserved weak hypercharge whose initial value we assume to be zero. The models exhibit two further zero eigenmodes. The first is 
which has zero eigenvalue (is conserved) in all boson interactions. A second zero eigenmode, 
is present if scalar-fermion interactions are removed. (3) (termed ``fiveness'') corresponds to the net number density of the fermion species appearing in the representation. A density 
generated through Higgs decays would be distributed as 
through -conserving interactions. may be destroyed through exchanges of light Higgs bosons. A convenient choice of independent combinations of fermion densities is 
and 
mass generated in the minimal SU(5) model in which
the Yukawa
coupling is . Results are for 
GeV. The nonconservation
parameter 
is unknown but less than 1. (b) Evolution of independent quantum
number
densities as a function of temperature in the minimal SU(5) model.
denotes the net baryon number; 
the asymmetry between 
and 
densities; and 
the total asymmetry between fermions in the and 
representations of
SU(5); 1
TTeV
In these graphs the parameter has been scaled out. The dashed curves are results obtained by neglecting
light Higgs
boson exchange processes. For model I, according to the estimate for 
given above, an adequate baryon-number asymmetry will be generated only if 
as would be the case if very heavy fermions exist with masses 
Similar conclusions have been reached by Segre and Turner. (9) Figure 1(a) shows the baryon asymmetry (taking 
GeV and 
) as a function of 
for 
and 
obtained by numerically integrating the Boltzmann transport equations (1) and (2). When 
exchanges thermalize the produced in 
decay to the value 
meanwhile, is reduced by light Higgs boson interactions. The final attained is determined by the reduction in that occurs before exchanges cease to be important and becomes fixed. For 
the is not effective in destroying the baryon number built up through decay. The enhancement in the final value of around 
is a result of the transition between these two regions. The dashed curve shows the final baryon number if all interactions are artificially set to zero. Figure 1(b) shows the temperature development of the quantum number asymmetries 
and 
for the case 
with the solid (dashed) curves indicating the effect of including (excluding) the destruction of and by the interactions of the light Higgs doublet. The final is reduced from the value of 
given above due to the light Higgs boson exchanges.
For model II the final baryon number density as a function of 
is shown in Fig. 2 for different choices of 
Note that, when 
we have [assuming 
in the Born approximation] 
and hence no is generated. For 
the additional decay mode 
(where 
is a light Higgs boson) decreases the effective nonconservation, 
in 
decay. For 
and 
the final is negative and determined by vector thermalization of the positive produced in 
decay. For but 
the final baryon number is positive and determined mainly by inverse decays into 
The dominant term governing the time evolution of for 
is 
with similar equations for 
and 
Since 
and 
this term tends to drive positive. In general there are three linear combinations of 
and which decrease as exponentials until cut off at temperatures below 
The final value of thus depends sensitively on the initial values of 
and 
For this reason, it is adequate to assume that is produced and damped in successive independent stages as in simple models which treat only one quantum number. (2) , (10) For both 
and 
inverse decays into 
are no longer able to change the sign of the negative produced through decays and hence the final produced is negative. The possibility of changes in the sign of associated with detailed features of the boson spectrum indicates that no generic relation may be found between the definition of ``matter'' as given for the 
system and that determined from the cosmological baryon-number asymmetry.
as a function of
the mass for different choices of the mass. The results are for 
and 
GeV. The
nonconservation
parameter is unknown but less than 1. One of the authors (E.W.K.) wishes to thank G. Segre and M. Turner for conversations at the Aspen Center for Physics. We would also like to thank T. Goldman for discussions.
This work was supported in part by the U. S. Department of Energy under Contract No. DE-AC-03-79ER0068, by the Fleischman Foundation, and by the National Science Foundation under Grant No. PHY79-23638.
